Laboratory Calibrations of Fe xii–xiv Line-intensity Ratios for Electron Density Diagnostics

The radius of the electron beam, r_e, and the emission spectra of the ions were measured simultaneously using a high-resolution grazing-incidence grating spectrometer (HiGGS; Beiersdorfer et al. 2014a). The spectrometer has a grating with 2400 lines mm⁻¹, a radius of curvature of R = 44.3 m, and a liquid-nitrogen-cooled charge-coupled device (CCD) detector. No entrance slit was used in the setup. The allowed spectral lines observed in the extreme ultraviolet (EUV) are emitted from within the ∼60 μm wide electron beam (Marrs et al. 1995; Utter et al. 1999), thereby forming a narrow emission source that effectively serves as a slit (see, e.g., Liang et al. 2009a, 2009b).

We carried out EUV measurements in the ranges ≈185–205 and ≈255–276 Å. In order to accurately determine the intensity ratio of spectral lines, it is necessary to correctly identify the spectral lines and resolve any line blends. For our EUV measurements, we calibrated the wavelength range by using well-known, intense lines from ions of O and Fe. Wavelength listings from the National Institute of Standards and Technology (Kramida et al. 2018) and CHIANTI (Dere et al. 1997; Landi et al. 2013; Del Zanna et al. 2015), as well as our previous line identification experiments (Beiersdorfer et al. 2014b; Träbert et al. 2014b), were used to identify these and the other lines in our measured spectra. However, because of the complexity of the spectrum in this wavelength range, not all of the lines could be identified. The resolving power, λ/Δλ, of the HiGGS was in the range 2800–4000 for our experiments, where λ is the measured line center and Δλ is the line width. This is similar to that of spectrometers used for solar observations, such as EIS. Observed spectral lines may have blends from the same ion species or nearby charge states of the same ion. Since the Fe spectrum in our experiment is produced by a gas containing C and O atoms, blends with lines from those elements are also possible.

Figure 1 shows the spectrum obtained for E_e = 395 eV and I_e = 7 mA in the 185–205 Å spectral range. All of the identified strong lines and other lines of interest are listed in Table 1 with the tabulated wavelengths and transitions taken from CHIANTI. In this wavelength range, the spectrum is dense, and there were many blends. The measured FWHM of the lines was Δλ ≈ 0.050 Å, which is set by the electron beam width. This corresponds to a resolving power of λ/Δλ ≈ 4000. The magnification of the EUV spectrometer setup was 1.00 ± 0.05. The magnification is determined by the ratio of the distances of the grating from the detector and from the source. Here and throughout, all uncertainties are given for an estimated 1σ confidence level. The relative sensitivity of the detector is expected to be approximately constant across the ≈20 Å wavelength range. The CCD quantum efficiency may vary by 2%. Based on previous experience with similar gratings, the reflectance of the spectrometer grating, far away from any absorption edges, is expected to vary by less than 5% over 20 Å (e.g., May et al. 2003). Because the lines whose ratios we study are closer in wavelength, the uncertainties in the ratio are even smaller and are negligible compared to other uncertainty sources. Additionally, our results are consistent with a uniform sensitivity across the detector; that is, we do not find larger discrepancies with theory in the ratios of widely separated lines versus closely spaced lines.

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Table 1.  Observed Spectral Lines of Interest in Figure 1

Wavelength (Å)	Iona	Lower Level	Upper Level	Comments
185.21	Fe viii	$3{p}^{6}3d\,{}^{2}{D}_{5/2}$	$3{p}^{5}{(}^{2}{P}^{o})3{d}^{2}{(}^{3}F)\,{}^{2}{F}_{7/2}^{o}$
186.60	Fe viii	$3{p}^{6}3d\,{}^{2}{D}_{3/2}$	$3{p}^{5}{(}^{2}{P}^{o})3{d}^{2}{(}^{3}F)\,{}^{2}{F}_{5/2}^{o}$
186.75	U	⋯	⋯	Blended
186.80	U	⋯	⋯	''
186.85	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{2}P)3d\,{}^{2}{F}_{5/2}$	''
186.88	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{5/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{2}P)3d\,{}^{2}{F}_{7/2}$	''
188.19	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{1/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{2}P)3d\,{}^{2}{D}_{3/2}$	Blended
188.21	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{3}{P}_{2}$	$3{s}^{2}3{p}^{3}{(}^{2}{D}^{2})3d\,{}^{3}{P}_{2}^{o}$	''
188.30	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{3}{P}_{2}$	$3{s}^{2}3{p}^{3}{(}^{2}D)3d\,{}^{3}{P}_{2}$
190.98	U	⋯	⋯	Blended
191.04	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{3}P)3d\,{}^{2}{D}_{5/2}$	''
192.02	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{3}{P}_{1}$	$3{s}^{2}3{p}^{3}{(}^{2}{D}^{o})3d\,{}^{1}{P}_{1}^{o}$
192.29	U	⋯	⋯	Blended
192.34	U	⋯	⋯	''
192.39	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{4}{S}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{3}P)3d\,{}^{4}{P}_{1/2}$	''
192.63	Fe xiv	$3s3{p}^{2}\,{}^{2}{D}_{5/2}$	$3s3{p}^{3}{(}^{1}{P}^{o})3d\,{}^{2}{F}_{5/2}^{o}$
192.75	O v	$1{s}^{2}2s2p\,{}^{3}{P}_{0}^{o}$	$1{s}^{2}2s3d\,{}^{3}{D}_{1}$	Blended
192.82	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{3}{P}_{1}$	$3{s}^{2}3{p}^{3}{(}^{2}{D}^{0})3d\,{}^{3}{P}_{2}^{o}$	''
192.91	O v	$1{s}^{2}2s2p\,{}^{3}{P}_{2}^{o}$	$1{s}^{2}2s3d\,{}^{3}{D}_{3}$	''
193.42	Fe vi	$3{p}^{6}3{d}^{2}\,{}^{3}{P}_{2}$	$3{p}^{5}{(}^{2}{P}^{o})3{d}^{3}{(}^{2}{D}_{2}^{o})\,{}^{1}{D}_{2}$	Blended
193.46	U	⋯	⋯	''
193.51	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{4}{S}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{3}P)3d\,{}^{4}{P}_{3/2}$	''
195.02	U	⋯	⋯	Blended
195.06	U	⋯	⋯	''
195.12	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{4}{S}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{3}P)3d\,{}^{4}{P}_{5/2}$	''
195.18	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{3/2}$	$3{s}^{2}3{p}^{2}{(}^{1}D)3d\,{}^{2}{D}_{3/2}$	''
196.42	Fe vi	$3{p}^{6}3{d}^{2}\,{}^{3}{F}_{2}$	$3{p}^{5}{(}^{2}{P}^{o})3{d}^{3}{(}^{2}H)\,{}^{3}{G}_{4}^{o}$	Blended
195.47	U	⋯	⋯	''
196.53	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{1}{D}_{2}$	$3{s}^{2}3p3d\,{}^{1}{F}_{5/2}^{o}$	''
196.64	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{5/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{1}D)3d\,{}^{2}{D}_{5/2}$	''
196.92	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{5/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{1}D)3d\,{}^{2}{D}_{3/2}$
197.02	C v	$1s2p\,{}^{1}{P}_{1}^{o}$	$1s4d\,{}^{1}{D}_{2}$
197.43	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{0}$	$3{s}^{2}3p3d\,{}^{3}{D}_{1}^{o}$
197.86	Fe ix	$3{s}^{2}3{p}^{5}3d\,{}^{1}{P}_{1}$	$3{s}^{2}3{p}^{5}4p\,{}^{1}{S}_{0}$
199.90	U	⋯	⋯	Blended
199.96	U	⋯	⋯	''
200.02	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{1}$	$3{s}^{2}3p3d\,{}^{3}{D}_{2}^{o}$	''
200.36	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{1}D)3d\,{}^{2}{S}_{1/2}$
201.11	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{3}{P}_{2}$	$3{s}^{2}3{p}^{3}{(}^{2}P)3d\,{}^{3}{D}_{3}$	Blended
201.13	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{1}$	$3{s}^{2}3p3d\,{}^{3}{D}_{1}^{o}$	''
201.14	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{3/2}^{o}$	$3{s}^{2}3{p}^{2}{(}^{3}P)3d\,{}^{2}{P}_{3/2}$	''
201.73	Fe xi	$3{s}^{2}3{p}^{4}\,{}^{1}{D}_{2}$	$3{s}^{2}3{p}^{3}{(}^{2}{D}^{o})3d\,{}^{1}{P}_{1}^{o}$	Blended
201.74	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{1/2}$	$3{s}^{2}3{p}^{2}3d\,{}^{2}{P}_{1/2}$	''
201.76	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{P}_{3/2}$	$3{s}^{2}3{p}^{2}3d\,{}^{2}{S}_{1/2}$	''
201.77	U	⋯	⋯	''
201.92	U	⋯	⋯	Blended
201.98	U	⋯	⋯	''
202.04	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{0}$	$3{s}^{2}3p3d\,{}^{3}{P}_{1}^{o}$	''
203.17	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{1}$	$3{s}^{2}3p3d\,{}^{3}{P}_{0}$
203.61	U	⋯	⋯	Blended
203.66	U	⋯	⋯	''
203.73	Fe xii	$3{s}^{2}3{p}^{3}\,{}^{2}{D}_{5/2}$	$3{s}^{2}3{p}^{2}{(}^{1}S)3d\,{}^{2}{D}_{5/2}$	''
203.77	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{D}_{1}$	$3{s}^{2}3p3d\,{}^{3}{F}_{2}$	''
203.79	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{2}$	$3{s}^{2}3p3d\,{}^{3}{D}_{2}^{o}$	''
203.83	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{2}$	$3{s}^{2}3p3d\,{}^{3}{D}_{3}^{o}$	''
204.26	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{1}$	$3{s}^{2}3p3d\,{}^{1}{D}_{2}^{o}$
204.93	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{3}{P}_{2}$	$3{s}^{2}3p3d\,{}^{3}{D}_{1}^{o}$

Notes. Wavelengths and transitions of the identified lines are from CHIANTI (Dere et al. 1997; Landi et al. 2013; Del Zanna et al. 2015).

^aU—Unidentified spectral line.

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Figure 2 shows the spectrum for the 255–276 Å range at the same beam energy and current as Figure 1. The identified transitions of interest are listed in Table 2. In contrast to the shorter wavelength range, this region has spectral features that are less complicated, and we were able to identify all of the important lines. The measured line width was ≈0.095 Å at a magnification of 1.03 ± 0.05 for this wavelength range. This corresponds to a resolving power of 2800. We did not perform experiments at the 475 eV beam energy for this spectral range, because our earlier experience with the shorter wavelength range showed no significant difference in the electron-density-diagnostic calibrations as the electron beam energy was varied.

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Table 2.  Observed Spectral Lines of Interest in Figure 2

Wavelength (Å)	Ion	Lower Level	Upper Level	Comments
256.38	Fe x	$3{s}^{2}3{p}^{5}\,{}^{2}{P}_{3/2}^{o}$	$3{s}^{2}3{p}^{4}{(}^{3}P)3d\,{}^{4}{D}_{3/2}$	Blended
256.42	Fe xiii	$3{s}^{2}3{p}^{2}\,{}^{1}{D}_{2}$	$3s3{p}^{3}\,{}^{1}{P}_{1}^{o}$	''
257.26	Fe x	$3{s}^{2}3{p}^{5}\,{}^{2}{P}_{3/2}^{o}$	$3{s}^{2}3{p}^{4}{(}^{3}P)3d\,{}^{4}{D}_{7/2}$	Blended
257.39	Fe xiv	$3{s}^{2}3p\,{}^{2}{P}_{1/2}^{o}$	$3s3{p}^{2}\,{}^{2}{P}_{1/2}$	''
264.79	Fe xiv	$3{s}^{2}3p\,{}^{2}{P}_{3/2}^{o}$	$3s3{p}^{2}\,{}^{2}{P}_{3/2}$
270.51	Fe xiv	$3{s}^{2}3p\,{}^{2}{P}_{3/2}^{o}$	$3s3{p}^{2}\,{}^{2}{P}_{1/2}$
274.20	Fe xiv	$3{s}^{2}3p\,{}^{2}{P}_{1/2}^{o}$	$3s3{p}^{2}\,{}^{2}{S}_{1/2}$

Note. Wavelengths and transitions are from CHIANTI (Dere et al. 1997; Landi et al. 2013; Del Zanna et al. 2015).

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We fit the observed spectra with Gaussian line profiles in order to extract the intensities of the various lines of interest, separate some line blends, and quantify the line widths. For these fits, the Gaussian widths were constrained to be equal, because those widths are set by the width of the electron beam and so should be the same for all unblended lines. The background level of the CCD was determined from the two-dimensional images, as described in Arthanayaka et al. (2018). This enabled us to set the background level to zero for the collapsed one-dimensional data on which we performed the fitting. In the least-squares fitting routine, we allowed the centroids to vary freely. The line-fitting scheme for each set of lines in a given wavelength range was kept consistent for different beam currents.

This fitting scheme is subject to some systematic error, and other reasonable fitting schemes could have been chosen. For example, the background subtraction we performed is imperfect, so we could incorporate additional free parameters to account for the background level, such as a constant or a linear term. In principle, all of the observed lines are expected to have the same width. However, unknown blends might cause minor broadening of the lines. To account for that, we could allow the widths of all the Gaussian components to be independent. We have tested the effects of these various fitting schemes and found that they cause the intensity ratio to vary by about ±8%.

Figures 3 and 4 illustrate example fits for E_e = 395 eV and I_e = 7 mA in the 185–205 and 255–276 Å wavelength ranges, respectively. The corresponding extracted spectral line intensities are listed in Tables 3–5.

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Intensities measured in EBIT can sometimes differ from theoretical predictions or astrophysical observations because of polarization effects. These effects arise because the ions in EBIT are excited by a directed electron beam. The resulting emission may not be isotropic, but we measure it only from a direction perpendicular to the electron beam axis. In contrast, theoretical calculations model the total intensity over all 4π solid angles. Moreover, most natural systems do not have such a preferred axis as exists in the experiment. The effect of this anisotropic emission is quantified by the polarization P given by

$$\begin{eqnarray}&&P=\displaystyle \frac{{I}_{\parallel }-{I}_{\perp }}{{I}_{\parallel }+{I}_{\perp }},\end{eqnarray} \tag{ 1 }$$

where I_∥ and I_⊥ represent the line intensities emitted parallel and perpendicular to the electron beam axis. In the experiment, we observe at approximately 90° to the axis of the beam and denote the measured intensity as I₉₀. This measured intensity is related to the total intensity I = I_∥ + I_⊥ emitted over all 4π solid angles by

$$\begin{eqnarray}&&{I}_{90}=\displaystyle \frac{3I}{3-P}.\end{eqnarray} \tag{ 2 }$$

Liang et al. (2009a) have calculated the polarization for many of the lines we have measured. For most of these lines, the polarization is only a few percent, and the effects on the intensity are negligible and can be ignored. In addition, one needs to consider that the electrons in an EBIT spiral around the magnetic field lines with a pitch angle, θ₀, measured relative to the field lines, that is given by $\epsilon ={\sin }^{2}{\theta }_{0}={E}_{\perp }/{E}_{{\rm{e}}}$ (Gu et al. 1999b). The resulting reduction in the polarization for electric dipole transitions is

$$\begin{eqnarray}&&{P}^{{\prime} }=P\displaystyle \frac{2-3\epsilon }{2-\epsilon P}.\end{eqnarray} \tag{ 3 }$$

In EBIT-I, E_⊥ ∼ 50–200 eV (Levine et al. 1989; Beiersdorfer et al. 1992; Gu et al. 1999b; Beiersdorfer & Slater 2001). The largest predicted polarization here is for the Fe xiii 202.04 Å line with P = 21%. Taking the spiraling of the electrons into account reduces this to 6%–18% for the beam energies considered here. This is discussed in more detail in Section 5.2.

Visible lines from metastable ions were used to determine the size of the trapped ion cloud. Because visible light arises from levels that have long lifetimes of ∼10⁻³ s, the metastable ion emission spans the entire ion cloud and gives an accurate measurement of its dimensions. Since we are concerned with density-diagnostic measurements for Fe xii, xiii, and xiv, it would have been preferable to measure the individual ion cloud sizes for each charge state. However, the visible lines from metastable Fe xii and xiii were too weak to be detected. We were only able to measure the metastable Fe xiv $3{s}^{2}\,3p{}^{2}{P}_{3/2}-3{s}^{2}\,3p{}^{2}{P}_{1/2}$ transition at 5302.9 Å. This emission was isolated by using a 30 Å bandpass filter centered at 5320 Å. The lifetime of the Fe xiv 5302.9 Å transition is 16.7 ms (Beiersdorfer et al. 2003; Brenner et al. 2007), which is long enough that the emission spans the ion cloud.

The measurement of the Fe xiv ion cloud diameter is expected to provide a reasonable approximation for nearby charge states. This can be understood in terms of the electrostatic and magnetic trapping that constrains the ion orbits. For electrostatic trapping, the maximum extent of the ion orbits is set by the balance between the ion kinetic energy and the ion charge q multiplied by the electrostatic potential ϕ. The ion temperatures in EBIT-I have been estimated to be ≈10q eV (Beiersdorfer et al. 1995). However, the potential energy depth of the trap, q ϕ, is also proportional to q. Consequently, for electrostatic trapping of a given element, the ion cloud radius should be independent of charge since the potential and kinetic energies of the ions are both proportional to q, and so the radial turning points of their orbits are all the same. There is also magnetic trapping, in which the ion orbit radius is roughly given by the gyroradius, mv/qB, where m is the ion mass, v is the ion velocity, and B is the magnetic field strength. For trapped ions, the potential energy balances the kinetic energy so that the kinetic energy is ∝q ϕ, which implies that $v\propto \sqrt{q}$, from which it follows that the gyroradius is $\propto 1/\sqrt{q}$. Thus, for magnetic trapping, the ratio of the ion orbits from Fe xii and xiv is $\sqrt{13/11}\approx 1.09$, and we expect less than a 10% difference between the Fe xii, xiii, and xiv ion cloud radii.

The apparatus and procedure for measuring the ion cloud have been previously reported by Arthanayaka et al. (2018), but the earlier setup suffered from some optical aberrations, so we have made several improvements. To enhance the image quality, we have replaced the 2 inch diameter convex lens with a 4 inch diameter lens with high optical quality. Furthermore, a 1 inch diameter aperture was placed directly in front of the lens to minimize spherical aberration. Additional precautions were taken to minimize internal reflections of the optical system. The bandpass filter was placed between the EBIT-I window and the convex lens. For our setup, the ion cloud was measured with a magnification of 3.05 ± 0.06, where the magnification was determined based on the properties of the lens and the measured distances in the optical assembly.

Figure 5 shows a line-out of an ion cloud measurement, which was obtained for E_e = 395 eV and I_e = 7 mA. In order to calculate the FWHM of the ion cloud, Γ_i, the horizontal scale in pixels was converted to physical units based on the optical setup magnification and the pixel spacing of the CCD (see Arthanayaka et al. 2018). The ion cloud profile is dominated by a narrow Gaussian with an FWHM measured to be Γ₁ ≈ 110 μm (Tables 6 and 7). The second Gaussian component has a much smaller amplitude that is only ≈10% of the amplitude of the narrow component, but it is much broader with an FWHM of Γ₂ ≈ 400 μm. There was some question as to whether the low-amplitude, broad component is real or a systematic error due to residual optical aberrations. Our experience has shown that improvements to the optical system have substantially reduced the amplitude of the broad component, but it was not possible to eliminate the broad component entirely. The existence of these two components is supported by our modeling of the ion cloud by calculating ion trajectories in EBIT-I. As discussed in Section 3.2, our model for the ion cloud qualitatively reproduces both the narrow and broad component structures of the observed ion cloud. For these reasons, we believe that the broad component is real and contains about 30% of the ions. We found no systematic variations versus beam energy or beam current for either Gaussian component.

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Table 6.  Electron Cloud Width, Γ_e; Ion Cloud Component Amplitudes, ${A}_{\mathrm{1,2}}$, and Widths ${{\rm{\Gamma }}}_{\mathrm{1,2}};$ Nominal Beam Density, ${\bar{n}}_{{\rm{e}}};$ and Effective Density, n_eff, for E_e = 395 eV

Beam Current	Γe	A1	Γ1	A2	${{\rm{\Gamma }}}_{2}$	${\bar{n}}_{{\rm{e}}}$	neff
(mA)	(μm)	(counts)	(μm)	(counts)	(μm)	(1011 cm−3)	(1011 cm−3)
1	56.4 ± 7.4	1958 ± 88	114.6 ± 5.6	0	0	0.74	0.288 ± 0.027
2	55.6 ± 7.1	4547 ± 157	93.5 ± 3.8	545 ± 127	392 ± 84	1.53	0.551 ± 0.073
3	60.0 ± 8.2	3067 ± 113	107.8 ± 4.4	904 ± 107	383 ± 29	1.98	0.505 ± 0.050
5	58.1 ± 7.1	3854 ± 153	103.9 ± 4.5	385 ± 143	416 ± 78	3.55	1.25 ± 0.17
7	58.6 ± 7.4	2659 ± 62	108.6 ± 2.7	432 ± 58	375 ± 33	4.94	1.51 ± 0.12
8	60.4 ± 8.6	3265 ± 74	108.1 ± 2.8	551 ± 67	401 ± 33	5.35	1.65 ± 0.15

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Table 7.  Same as Table 6 but for ${E}_{{\rm{e}}}=475\,\mathrm{eV}$

Beam Current	Γe	A1	Γ1	A2	Γ2	${\bar{n}}_{{\rm{e}}}$	neff
(mA)	(μm)	(counts)	(μm)	(counts)	(μm)	$({10}^{11}\ {\mathrm{cm}}^{-3})$	$({10}^{11}\ {\mathrm{cm}}^{-3})$
1	51.2 ± 6.3	4578 ± 152	94.3 ± 3.7	537 ± 121	414 ± 91	0.082	0.254 ± 0.033
2	53.3 ± 7.2	970 ± 174	123 ± 13	169 ± 123	336 ± 107	1.51	0.348 ± 0.094
3	56.7 ± 6.7	2797 ± 447	103 ± 11	751 ± 263	267 ± 68	2.02	0.63 ± 0.14
5	57.0 ± 6.3	2337 ± 150	102.1 ± 7.2	496 ± 141	357 ± 11	3.35	0.98 ± 0.14
7	54.4 ± 6.1	1663 ± 78	110.4 ± 5.4	476 ± 72	374 ± 38	5.18	1.13 ± 0.12
9	57.4 ± 6.5	1883 ± 73	107.7 ± 5.1	419 ± 60	462 ± 50	6.04	1.46 ± 0.16

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The electron beam is nearly Gaussian, with the density varying radially as

$$\begin{eqnarray}&&{n}_{{\rm{e}}}(r)={n}_{0}{e}^{-{r}^{2}/2{\sigma }_{{\rm{e}}}^{2}},\end{eqnarray} \tag{ 4 }$$

where n₀ is the central density of the beam and σ_e is the Gaussian beam width, and the FWHM Γ_e = 2.355σ_e. It is useful to characterize the electron beam density using an average value, which we take to be the density that the beam would have if it were a uniform cylindrical electron beam of radius r_e. We refer to this as the geometric average, denoted by the symbol ${\bar{n}}_{{\rm{e}}}$.

For a uniform cylindrical electron beam, ${\bar{n}}_{{\rm{e}}}$ is related to the electron current, I_e, by

$$\begin{eqnarray}&&{\bar{n}}_{{\rm{e}}}=\displaystyle \frac{{I}_{{\rm{e}}}}{\pi {r}_{{\rm{e}}}^{2}{{ev}}_{{\rm{e}}}},\end{eqnarray} \tag{ 5 }$$

where r_e is the radius of the beam and v_e is the axial velocity of the electrons. Experimentally, I_e is measured by an ammeter connected to the EBIT collector, and v_e is calculated from the space-charge-corrected electron beam energy E_e.

There remains a choice as to how we should define r_e for the electron beam and the corresponding equivalent cylindrical beam from which we derive our average. Here, we choose to define r_e = 2σ_e so that 95% of the electrons in the beam are within this radius. Alternative conventions are also possible. For example, Liang et al. (2009b) defined their beam radius to be the radius that enclosed 80% of the beam electrons. Our choice of r_e = 2σ_e has the property that the central density n₀ is twice the geometric average density ${\bar{n}}_{{\rm{e}}}$. Moreover, the density of the Gaussian beam at r_e is approximately one-fourth of ${\bar{n}}_{{\rm{e}}}$. Thus, we can characterize our beam density by ${\bar{n}}_{{\rm{e}}}$.

The measurements of the geometry of the electron beam have been described in Arthanayaka et al. (2018). Briefly, we use the measured spectroscopic EUV line shapes to infer the electron beam profile. This works because the EUV emission comes from allowed transitions that are excited and decay within the electron beam. The typical lifetimes of the transitions are 10⁻¹⁰ s, and the thermal energies of the ions in EBIT are estimated to be ∼10q ∼ 10² eV, which corresponds to a velocity of 3 × 10⁴ m s⁻¹. For a typical 60 μm electron beam width, the crossing time of an iron ion is about 10⁻⁹ s, which is much shorter than the lifetimes of the EUV-radiating levels. Doppler broadening of the spectral lines is small and adds in quadrature with the broadening, due to the finite width of the emission source. As a result, the widths of the EUV spectral lines reflect mainly the width of the electron beam. As these contributions add in quadrature, Doppler broadening contributes up to 10% of the line widths, which we incorporate into the analysis as a 10% uncertainty in the electron beam width. Gaussian profiles were fit to the EUV spectral lines in order to infer the electron beam FWHM Γ_e. For these fits, the x-axis of the spectra was calibrated in physical units via the pixel size, rather than in wavelength units. To determine the beam size, strong lines were used that were less affected by blends, as described in Arthanayaka et al. (2018). The beam width was derived for each beam current and energy studied. There was some slight variation as a function of current and energy, with values in the range Γ_e ≈ 50–60 μm (Tables 6 and 7). This is consistent with previous measurements on EBIT-I (Marrs et al. 1995; Utter et al. 1999).

For a two-level model, let f be the relative population of the upper level so that 1 − f is the population of the lower level. Also, n_e is the electron density, C is the collisional excitation rate coefficient, D is the collisional de-excitation rate coefficient, and A_r is the radiative de-excitation rate. Then the differential equation describing the level population is

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{{dt}}={n}_{{\rm{e}}}C(1-f)-{n}_{{\rm{e}}}{Df}-{A}_{r}f.\end{eqnarray} \tag{ 8 }$$

This equation can be solved analytically. The equilibrium level is given by df/dt = 0, so

$$\begin{eqnarray}&&{f}_{\mathrm{eq}}=\displaystyle \frac{{n}_{{\rm{e}}}C}{{n}_{{\rm{e}}}(C+D)+{A}_{r}}.\end{eqnarray} \tag{ 9 }$$

Defining an initial condition that f(t = 0) = f₀, the solution to Equation (8) is

$$\begin{eqnarray}\begin{array}{rcl}f(t) & = & \displaystyle \frac{{n}_{{\rm{e}}}C}{{n}_{{\rm{e}}}(C+D)+{A}_{r}}\left(1-{e}^{-[{n}_{{\rm{e}}}(C+D)+{A}_{r}]t}\right)\\ & & +\,{f}_{0}{e}^{-[{n}_{{\rm{e}}}(C+D)+{A}_{r}]t}.\end{array}\end{eqnarray} \tag{ 10 }$$

For the low-density case, collisions are negligible, so

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{{dt}}=-{A}_{r}f.\end{eqnarray} \tag{ 11 }$$

The equilibrium solution for the low-density case is f = 0, indicating that all the ions are in the ground state. As a function of time, the evolution to this "coronal limit" is given by

$$\begin{eqnarray}&&f(t)={f}_{0}{e}^{-{A}_{r}t}.\end{eqnarray} \tag{ 12 }$$

Now, consider an ion passing in and out of the beam and starting in the low-density equilibrium with f(0) = 0. Suppose the ion passes into the beam for a time Δt_hi and then is outside the beam for a time Δt_lo, with the sequence repeating many times. Let the index i refer to the time step for completing the entire sequence so that ${t}_{i}=i({\rm{\Delta }}{t}_{\mathrm{lo}}+{\rm{\Delta }}{t}_{\mathrm{hi}})$. It is also convenient to introduce a shorthand for the exponentials, so we set $\eta ={e}^{-[{n}_{{\rm{e}}}(C+D)+{A}_{r}]{\rm{\Delta }}{t}_{\mathrm{hi}}}$ and $\lambda ={e}^{-{A}_{r}{\rm{\Delta }}{t}_{\mathrm{lo}}}$. At each time step, the level populations are given by

$$\begin{eqnarray}&&{f}_{i}=\displaystyle \frac{{n}_{{\rm{e}}}C}{{n}_{{\rm{e}}}(C+D)+{A}_{r}}(1-\eta )\lambda +\eta \lambda {f}_{i-1}.\end{eqnarray} \tag{ 13 }$$

In equilibrium it must be that ${f}_{i}={f}_{i-1}={f}_{\infty }$. Substituting into Equation (13) and solving for f_∞, we find that

$$\begin{eqnarray}&&{f}_{\infty }=\displaystyle \frac{{n}_{{\rm{e}}}C}{{n}_{{\rm{e}}}(C+D)+{A}_{r}}\displaystyle \frac{(1-\eta )\lambda }{1-\eta \lambda }.\end{eqnarray} \tag{ 14 }$$

Since the population change at each ${\rm{\Delta }}{t}_{\mathrm{hi},\mathrm{lo}}$ is very small, we can use a Taylor series expansion for the exponential terms so that $\eta \approx 1-[n(C+D)+A]{\rm{\Delta }}{t}_{\mathrm{hi}}$ and $\lambda \approx 1-A{\rm{\Delta }}{t}_{\mathrm{lo}}$. Substituting these into Equation (14) and eliminating second-order terms give

$$\begin{eqnarray}&&{f}_{\infty }=\displaystyle \frac{{n}_{{\rm{e}}}C}{{n}_{{\rm{e}}}(C+D)+A}\displaystyle \frac{[{n}_{{\rm{e}}}(C+D)+A]{\rm{\Delta }}{t}_{\mathrm{hi}}}{[{n}_{{\rm{e}}}(C+D)+A]{\rm{\Delta }}{t}_{\mathrm{hi}}+A{\rm{\Delta }}{t}_{\mathrm{lo}}}.\end{eqnarray} \tag{ 15 }$$

This is the high-density equilibrium population ${n}_{{\rm{e}}}C/[{n}_{{\rm{e}}}(C\,+D)+{A}_{r}]$ multiplied by a correction factor.

The effective density is defined as the time-averaged density experienced by the ion. In this example, the ion travels from the high-density electron beam with density n_e for time Δt_hi and then is outside the beam where the density is 0 for a time Δt_lo. So we have

$$\begin{eqnarray}&&{n}_{\mathrm{eff}}=\displaystyle \frac{{n}_{{\rm{e}}}{\rm{\Delta }}{t}_{\mathrm{hi}}}{{\rm{\Delta }}{t}_{\mathrm{hi}}+{\rm{\Delta }}{t}_{\mathrm{lo}}}.\ \end{eqnarray} \tag{ 16 }$$

Rewriting Equation (15) in terms of n_eff gives

$$\begin{eqnarray}&&{f}_{\infty }=\displaystyle \frac{{n}_{\mathrm{eff}}C}{{n}_{\mathrm{eff}}(C+D)+A}.\end{eqnarray} \tag{ 17 }$$

But this is the equilibrium level population if the ions are always in a region where the electron density is n_eff. This demonstrates that the effective density controls the level populations.

We have calculated the ion orbit trajectories in order to determine the amount of time that ions spend inside and outside the beam and to better understand the observed shape of the ion cloud. The trajectories are calculated in the two-dimensional plane perpendicular to the beam axis. This is justified by the axial symmetry of EBIT. The electron beam is modeled as a Gaussian distribution, as given by Equation (4), with n₀ and σ_e set by the experimentally measured values. The electron beam produces an electric-field vector, which can be determined analytically at any location. The experiment also has an axial 3 T magnetic field, which is included in the model. Ion trajectories were calculated using test particles by integrating the equation of motion for the Lorentz force from the combined axial magnetic field and radial electric field. As a result, ion space-charge effects are ignored, but these are expected to be small since the density of ions is low. The integration was carried out using the Boris algorithm (Birdsall & Langdon 2005; Qin et al. 2013). The results were tested for energy conservation and found to conserve energy to about a few percent over at least 10 ion cyclotron periods. We compare to this timescale because it is roughly similar to the timescales for the radial oscillations of the ions. The method is very fast, allowing for 1000 ion trajectories to be calculated in minutes.

Test particle Fe¹³⁺ ions were simulated with experimentally motivated initial conditions. The ion temperature in EBIT is estimated to be about 10q eV (Beiersdorfer et al. 1995). Based on those measurements, the initial velocities for Fe¹³⁺ were drawn at random from a Maxwellian distribution with a temperature of 130 eV. The initial positions were assumed to be proportional to the local electron density, as we expect the formation rate to be proportional to the electron beam density profile. For each simulated ion trajectory, we calculated the fraction of time that the ion spent at $r\lt {r}_{{\rm{e}}}=2{\sigma }_{{\rm{e}}}$, which we consider to be the time that the ion is "in the beam." We found that the ion trajectories typically spend about 40% of their time in the electron beam. This is roughly consistent with our finding that the effective density is about 30% of the average electron beam density.

In order to simulate the ion cloud measurement, we chose a late time in the simulation so that the orbits had time to become incoherent and spread out from their initial positions, which were mainly within the electron beam. At that time, we made a histogram of the number of ions that would be seen along a line of sight looking through the device. In practice, this was done by making a histogram of the x-coordinates of the ions, rather than their radius r. This is the Abel transform of the radial distribution. We found that our simulated ion clouds could be described by two Gaussian components: a strong, narrow component with Γ ≈ 60 μm and a weaker, broad component with Γ ≈ 300 μm. The amplitude of the broad component is about 10% that of the narrow component, and it contains about 30% of the ions. These characteristics are very similar to the measured ion cloud. One difference with the experiment is that both components in the simulated distribution have slightly smaller widths than in the experiment, which suggests that some ions in the model are more strongly confined to the electron beam than they are in the experiment. Although we do not reproduce the precise widths of the observed ion clouds, it is interesting that the model reproduces the general structure with roughly similar values for the widths of both the narrow and broad components and similar amplitudes. These results support the existence of the broad component seen in the experiment.

A possible explanation for the differences between the simulated versus measured ion clouds is that the ion velocity distribution that we use to initialize the ions in the simulation could be inaccurate. In reality, the ions could have a greater temperature than we assumed or they could have a nonthermal distribution. Alternatively, there could be collisions in the electron beam that tend to scatter the ions to wider orbits.

Fe xii has several bright, unblended lines that are density insensitive with respect to one another at 192.39, 193.51, and 195.12 Å. For this reason, these lines are used to calibrate instruments and test theoretical models (Del Zanna & Mason 2005). We have used ratios among these lines for a similar purpose, to ensure the consistency of our results. Figure 6 demonstrates that there is no significant density sensitivity for ratios among these lines and that the predicted ratios are in good agreement with the calculations. The weighted value for our measurement of the 192.39/195.12 ratio is 0.68 ± 0.02, and the weighted mean for the 193.51/195.12 ratio was 0.32 ± 0.01. These may be compared to the theoretically predicted values of 0.67 and 0.32, respectively. Good agreement was also found for this line ratio in the earlier EBIT measurements of Träbert et al. (2014a), though the density was not determined for those measurements.

Density diagnostics for Fe xii can be formed by taking the ratio of a density-sensitive line with any of the above three density-insensitive lines (Flower 1977; Dere et al. 1979; Schmitt et al. 1996; Del Zanna & Mason 2005; Young et al. 2007). Two Fe xii density-sensitive lines were observed in the measured spectral range: the $3{s}^{2}\,3{p}^{3}{}^{2}{D}_{5/2}^{o}-3{s}^{2}\,3{p}^{2}{(}^{2}P)3d{}^{2}{F}_{7/2}$ transition at 186.88 Å and the $3{s}^{2}\,3{p}^{3}{}^{2}{D}_{5/2}^{o}-3{s}^{2}\,3{p}^{2}{(}^{1}D)3d{}^{2}{D}_{5/2}$ transition at 196.64 Å. The 186.88 Å line is blended with the Fe xii $3{s}^{2}\,3{p}^{3}{}^{2}{D}_{3/2}^{o}-3{s}^{2}\,3{p}^{2}{(}^{2}P)3d{}^{2}{F}_{5/2}$ transition at 186.85 Å. Because the blend is difficult to separate reliably, we study the ratio of the total intensity of both (186.85 + 186.88) Å lines relative to the 195.12 Å line. The same procedure is commonly used in the analysis of solar observations. The Fe xii line at 196.64 Å is close to an Fe xiii line at 196.53 Å, but our spectral resolution is sufficient to clearly separate these lines, as can be seen in Figure 3(e). We note that the spectral resolution of EIS is similar to that of our experiment, so the 196 Å blend is also separable in solar observations.

Our results for these two Fe xii ratios are shown in Figure 7. The intensity ratio for the 196.64/195.12 lines is in reasonable agreement with the predicted curve. For the (186.85 + 186.88)/195.12 ratio, the ratio versus density curve is significantly different from the calculation, with the measured ratio having a greater magnitude and a steeper slope versus density than predicted. The difference is above the estimated 20% uncertainty for the calculations. The polarization for these lines is only a few percent (Liang et al. 2009a), and polarization effects cannot account for the observed discrepancy with theory.

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When using the intensity ratio as a diagnostic, the inferred densities change exponentially as a function of the ratio, so the inferred density is very sensitive to small changes in the ratios. For the (186.85 + 186.88)/195.12 diagnostic, the interpretation of observed line intensities based on the current theoretical values would overestimate the density by about an order of magnitude.

Young et al. (2009) discussed these Fe xii line ratios, (186.85 + 186.88)/195.12 and 196.64/195.12, in the solar spectrum and found that the derived densities disagreed with one another by at least a factor of 2. Based on our results, we believe this difference is due to theoretical uncertainties in the (186.85+186.88)/195.12 diagnostic, which appears to be much larger than the 20% often quoted for the uncertainties in theoretical values of radiative transition and collisional excitation rates. This underscores the difficulty of estimating errors in theoretical atomic data for such complex ions and the importance of laboratory measurements for benchmarking the calculations.

We were able to identify one density-insensitive Fe xiii line ratio involving the $3{s}^{2}\,3{p}^{2}{}^{3}{P}_{0}-3{s}^{2}\,3p\,3d{}^{3}{D}_{1}^{o}$ and $3{s}^{2}3{p}^{2}{}^{3}{P}_{2}-3{s}^{2}\,3p\,3d{}^{3}{D}_{1}^{o}$ transitions at 197.43 and 204.93 Å, respectively (Del Zanna 2012). These lines are relatively weak compared to the other spectral lines that we are interested in, but they could be measured reliably for higher electron beam currents. Figure 8 plots the measured ratios for these lines and compares them to the predictions of FAC. The ratio is indeed density insensitive. However, we found that the ratio disagreed with the FAC calculations. The weighted mean of our measurements is 0.53 ± 0.01.

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Initially, the theoretical calculations were performed with limited configuration interaction that included only states with one electron in the 3d orbital, and they produced a predicted ratio of ∼1. We recalculated the ratio by including configurations with double excitation to the 3d orbitals. This reduced the predicted ratio to ∼0.82 and is close to the value in CHIANTI of about 0.86. Clearly, a large discrepancy still exists between theory and experiment. These two lines originate from the same upper level, so their intensity ratio is strictly that of the respective radiative decay rates and depends solely on the atomic structure model. It is possible that including even more configuration interaction in the model might resolve the discrepancy. However, including all double excitations to even higher orbitals results in extremely large Hamiltonian matrices, and the model quickly becomes computationally intractable. A more manageable solution is to use a many-body perturbation approach to evaluate the effect of further electron correlation effects. However, this is beyond the scope of the present study.

The intensity ratios of the Fe xiii lines within the 203.83 Å line blend (see Table 1) and the 202.04 Å line are plotted in Figure 9. For the densities accessible in the experiment, we are in the high-density limit of this diagnostic and observed little variation in the ratio as a function of density. The ratio itself was higher by about 25% than that predicted by the FAC calculations, though as we explain below this appears to lie within the combined experimental and theoretical uncertainties. The same line ratio in CHIANTI for a T = 5 MK Maxwellian agrees with our FAC calculations to within a few percent. Hence, the difference we find with experiment also exists in the CHIANTI data.

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We considered the effects of polarization as a possible explanation for this difference, but we find that polarization cannot account for the difference between theory and experiment. According to Liang et al. (2009a), the polarization of most of the Fe xiii lines that we have measured is only a few percent, and the effects on intensity are negligible. The Fe xiii 202.04 Å line, though, has a significant polarization of 21%. Spiraling of the electron beam reduces this to less than 18%, which is predicted to cause at most a 6% difference in the measured versus theoretical intensity. However, this polarization effect would imply that we have overestimated the intensity of the 202.04 Å line. As this line is in the denominator of our density diagnostic, correcting for this effect would actually exacerbate the difference with theory, rather than resolve it.

Blending issues in the experiment are another possible explanation for the disagreement with theory. The 203.83 Å lines form a very complex blend, and it is possible that there is a systematic overestimate of its intensity. The complex consists of six closely spaced lines: four that are identified as Fe XII 203.72 Å and Fe XIII 203.77, 203.80, and 203.83 Å, plus two other unidentified lines at 203.60 and 203.66 Å. Further, we have observed O V lines in this spectrum, so it is possible that the weak O v $1{s}^{2}\,2{p}^{2}{}^{3}{P}_{0}$–$1{s}^{2}\,2p(2{P}_{1/2}^{o})3d{}^{3}{D}_{1}^{o}$ and $1{s}^{2}\,2{p}^{2}{}^{3}{P}_{1}$–$1{s}^{2}\,2p(2{P}_{1/2}^{o})3d{}^{3}{D}_{1}^{o}$ transitions at 203.78 and 203.82 Å may also contribute to this complex. Given these experimental factors and the estimated 20% systematic uncertainty in the calculations, we estimate that the observed differences between theory and experiment for this ratio are within the combined uncertainties.

In Figure 9 we also plot the intensity ratios for the 196.63/202.04 and 200.02/202.04 density diagnostics. For the 196.53/202.04 ratio, the density diagnostic is not quite saturated, and we find that the experimental ratio is about 20% larger than the theoretical prediction, but within the combined experimental and theoretical uncertainties. Our measurements of the 200.02/202.04 diagnostic are in the high-density saturation limit of the diagnostic, and the experimental ratios are in good agreement with the calculations.

We have also measured density-sensitive Fe xiv lines in the wavelength range 255–277 Å (Malinovsky & Heroux 1973; Del Zanna et al. 2012). The spectra in this range are much cleaner, with fewer blends than in the short wavelength range containing the Fe xii and xiii diagnostics. However, the intensities of the lines were somewhat weaker and required longer exposure times. Because our results for Fe xii and xiii found that there was no line-ratio dependence seen between E_e = 395 and 475 eV, we carried out experiments only for E_e = 395 eV, enabling us to obtain longer-exposure spectra.

We measured the line-intensity ratios for 257.39/274.21, 264.79/274.21, and 270.52/274.21. The specific transitions are listed in Table 2. Our results are shown in Figure 10, where they are compared to the FAC calculations. For these ratios, we found reasonable agreement between these measured and predicted ratios. According to Del Zanna et al. (2012), the Fe xiv line at 264.79 Å is blended with an Fe xi spectral line, though the precise wavelength of that line is unknown. A blend contribution to the 264.79 Å line would systematically increase the measured line ratio relative to the prediction, which currently appears to be in excellent agreement with the experimental results. It is possible that the blend is present in solar spectra, but negligible in the experiment. However, we observe two Fe x lines in the spectrum (see Figure 1), which suggest that the Fe xi line should also be present in our spectrum. The fact that we do not notice any blending seems to indicate that the Fe xi line is either not at the predicted location or is weaker than calculated.

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Our measurements have identified several discrepancies with theoretical calculations. For Fe xii, the (186.85 + 186.88)/195.12 line ratio does not match the theoretical calculations. In particular, the measured ratio is larger, has a steeper density dependence, and has a greater value for the ratio in the high-density limit. A similar discrepancy exists for the ratio of the 186 Å blend with either the 193.51 or 192.39 Å lines, because we have confirmed that the 192.39, 193.51, and 195.12 Å lines are density insensitive with respect to one another.

For Fe xiii, a significant discrepancy was found for the density-insensitive ratio of the 197.43/204.94 lines. Our measurements confirm that the ratio is not sensitive to density, but the magnitude of the ratio differs by ≈30% from the value predicted by FAC or CHIANTI.

We measured three Fe xiii density diagnostics. Of these, the 200.02/202.04 line ratio appears to be the most accurate. Reasonable agreement between experiment and theory is also found for the 196.52/202.04 line ratio. For the (203.77 + 203.79 + 203.83)/202.04 diagnostic, our measurements are all in the high-density saturation limit of the diagnostic, where we found the measured ratio to be somewhat larger than predicted by theory. This diagnostic involves a complex blend that adds to the uncertainty in the measurement so that the disagreement is at the limit of the combined theoretical and experimental uncertainties.

There are two other EBIT measurements of the Fe xiii (203.77 + 203.79 + 203.83)/202.04 ratio. Liang et al. (2009a) measured this ratio at an estimated density of ≈3 × 10⁹ cm⁻³; while their measured value is in good agreement with the calculation, they did not directly measure the electron or ion beam profiles, but rather used a different line ratio to infer the electron density. Yamamoto et al. (2008) reported a measurement of this ratio from the Lawrence Livermore National Laboratory EBIT-II and again obtained good agreement at a density they estimated to be ≈2 × 10¹¹ cm⁻³, based on typical densities in EBIT-II and the inferred line ratios being in the high-density saturation limit. Two measurements using magnetic fusion devices with densities near 2 × 10¹³ cm⁻³ were also made using the Large Helical Device (Yamamoto et al. 2008) and the National Spherical Torus Experiment (Weller et al. 2018). Both of these produced values well below theory. However, they used spectrometers that did not resolve the extensive line blending with emission from neighboring charge states of iron.

Our measurements for several Fe xiv line ratios were all in agreement with theoretical calculations, though, again, our densities were near or above the high-density saturation of those diagnostics. The 264.79/274.21 and 270.52/274.21 diagnostics were also measured at lower densities by Nakamura et al. (2011), who found good agreement with atomic calculations. Based on our results and those of Nakamura et al., these Fe xiv diagnostics appear to be reliable.

For Fe xiv, there are also measurements on the National Spherical Torus Experiment at 2 × 10¹³ cm⁻³ (Weller et al. 2018) that agree well with the prediction for the 270.52/274.21 ratio, but are significantly larger than predicted for the 264.79/274.21 ratio. Although we do not expect line blending, these measurements were performed with much lower spectral resolution, and blending cannot be ruled out. Blending is also more likely in the tokamak experiments as those plasmas contain multiple other elements besides iron.